| L(s) = 1 | + 36·2-s + 784·4-s + 1.31e3·5-s − 4.48e3·7-s + 9.79e3·8-s + 4.73e4·10-s − 1.47e3·11-s − 1.51e5·13-s − 1.61e5·14-s − 4.88e4·16-s − 1.08e5·17-s + 5.93e5·19-s + 1.03e6·20-s − 5.31e4·22-s + 9.69e5·23-s − 2.26e5·25-s − 5.45e6·26-s − 3.51e6·28-s + 6.64e6·29-s + 7.07e6·31-s − 6.77e6·32-s − 3.89e6·34-s − 5.88e6·35-s − 7.47e6·37-s + 2.13e7·38-s + 1.28e7·40-s + 4.35e6·41-s + ⋯ |
| L(s) = 1 | + 1.59·2-s + 1.53·4-s + 0.940·5-s − 0.705·7-s + 0.845·8-s + 1.49·10-s − 0.0303·11-s − 1.47·13-s − 1.12·14-s − 0.186·16-s − 0.314·17-s + 1.04·19-s + 1.43·20-s − 0.0483·22-s + 0.722·23-s − 0.115·25-s − 2.34·26-s − 1.07·28-s + 1.74·29-s + 1.37·31-s − 1.14·32-s − 0.499·34-s − 0.663·35-s − 0.655·37-s + 1.66·38-s + 0.794·40-s + 0.240·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.348934505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.348934505\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 9 p^{2} T + p^{9} T^{2} \) |
| 5 | \( 1 - 1314 T + p^{9} T^{2} \) |
| 7 | \( 1 + 640 p T + p^{9} T^{2} \) |
| 11 | \( 1 + 1476 T + p^{9} T^{2} \) |
| 13 | \( 1 + 151522 T + p^{9} T^{2} \) |
| 17 | \( 1 + 108162 T + p^{9} T^{2} \) |
| 19 | \( 1 - 593084 T + p^{9} T^{2} \) |
| 23 | \( 1 - 969480 T + p^{9} T^{2} \) |
| 29 | \( 1 - 6642522 T + p^{9} T^{2} \) |
| 31 | \( 1 - 7070600 T + p^{9} T^{2} \) |
| 37 | \( 1 + 7472410 T + p^{9} T^{2} \) |
| 41 | \( 1 - 4350150 T + p^{9} T^{2} \) |
| 43 | \( 1 + 4358716 T + p^{9} T^{2} \) |
| 47 | \( 1 + 28309248 T + p^{9} T^{2} \) |
| 53 | \( 1 + 16111710 T + p^{9} T^{2} \) |
| 59 | \( 1 - 86075964 T + p^{9} T^{2} \) |
| 61 | \( 1 - 32213918 T + p^{9} T^{2} \) |
| 67 | \( 1 - 99531452 T + p^{9} T^{2} \) |
| 71 | \( 1 - 44170488 T + p^{9} T^{2} \) |
| 73 | \( 1 + 23560630 T + p^{9} T^{2} \) |
| 79 | \( 1 + 401754760 T + p^{9} T^{2} \) |
| 83 | \( 1 - 744528708 T + p^{9} T^{2} \) |
| 89 | \( 1 + 769871034 T + p^{9} T^{2} \) |
| 97 | \( 1 - 907130882 T + p^{9} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.60696650808100087458012565792, −17.51557137290252570088223125072, −15.81690395220482879536495864101, −14.33809375015404991495095619307, −13.29206482181447551758779020482, −12.03578742848102127369637849461, −9.833586942827917393792392911714, −6.62417014582556452549137957823, −5.03891167141543488409792631707, −2.76319651467288245806051970006,
2.76319651467288245806051970006, 5.03891167141543488409792631707, 6.62417014582556452549137957823, 9.833586942827917393792392911714, 12.03578742848102127369637849461, 13.29206482181447551758779020482, 14.33809375015404991495095619307, 15.81690395220482879536495864101, 17.51557137290252570088223125072, 19.60696650808100087458012565792
